# PERT distribution

Last updated
PERT
Probability density function
Example density curves for the PERT probability distribution
Cumulative distribution function
Example cumulative distribution curves for the PERT probability distribution
Parameters${\displaystyle b>a\,}$ (real)
${\displaystyle c>b\,}$ (real)
Support ${\displaystyle x\in [a,c]\,}$
PDF

${\displaystyle {\frac {(x-a)^{\alpha -1}(c-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )(c-a)^{\alpha +\beta -1}}}}$ where ${\displaystyle \alpha ={\frac {4b+c-5a}{c-a}}=1+4{\frac {b-a}{c-a}}}$

## Contents

${\displaystyle \beta ={\frac {5c-a-4b}{c-a}}=1+4{\frac {c-b}{c-a}}}$
CDF

${\displaystyle I_{z}(\alpha ,\beta )}$

(the incomplete beta function) with ${\displaystyle z=(x-a)/(c-a)}$
Mean ${\displaystyle \operatorname {E} [X]={\frac {a+4b+c}{6}}=\mu }$
Median

${\displaystyle I_{\frac {1}{2}}^{[-1]}(\alpha ,\beta )(c-a)+a}$

${\displaystyle \approx a+(c-a){\frac {\alpha -1/3}{\alpha +\beta -2/3}}={\frac {a+6b+c}{8}}}$
Mode ${\displaystyle b}$
Variance ${\displaystyle \operatorname {var} [X]={\frac {(\mu -a)(c-\mu )}{7}}}$
Skewness ${\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}}$
Ex. kurtosis ${\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}$

In probability and statistics, the PERT distribution is a family of continuous probability distributions defined by the minimum (a), most likely (b) and maximum (c) values that a variable can take. It is a transformation of the four-parameter Beta distribution with an additional assumption that its expected value is

${\displaystyle \mu ={\frac {a+4b+c}{6}}.}$

The mean of the distribution is therefore defined as the weighted average of the minimum, most likely and maximum values that the variable may take, with four times the weight applied to the most likely value. This assumption about the mean was first proposed in Clark, 1962 [1] for estimating the effect of uncertainty of task durations on the outcome of a project schedule being evaluated using the program evaluation and review technique, hence its name. The mathematics of the distribution resulted from the authors' desire to make the standard deviation equal to about 1/6th of the range. [2] [3] The PERT distribution is widely used in risk analysis [4] to represent the uncertainty of the value of some quantity where one is relying on subjective estimates, because the three parameters defining the distribution are intuitive to the estimator. The PERT distribution is featured in most simulation software tools.

## Comparison with the triangular distribution

The PERT distribution offers an alternative [5] to using the triangular distribution which takes the same three parameters. The PERT distribution has a smoother shape than the Triangular distribution. The triangular distribution has a mean equal to the average of the three parameters:

${\displaystyle \mu ={\frac {a+b+c}{3}}}$

The formula places equal emphasis on the extreme values which are usually less-well known than the most likely value, and can therefore be unduly influenced by poor estimation of an extreme. The triangular distribution also has an angular shape that does not match the smoother shape that typifies subjective knowledge:

## The modified-PERT distribution

The PERT distribution assigns very small probability to extreme values, particularly to the extreme furthest away from the most likely value if the distribution is strongly skewed. [6] [7] The Modified PERT distribution [8] was proposed to provide more control on how much probability is assigned to tail values of the distribution. The modified-PERT introduces a fourth parameter ${\displaystyle \gamma ,}$ that controls the weight of the most likely value in the determination of the mean:

${\displaystyle \mu ={\frac {a+\gamma b+c}{\gamma +2}}}$

Typically, values of between 2 and 3.5 are used for ${\displaystyle \gamma ,}$ and have the effect of flattening the density curve. This is useful for highly skewed distributions where the distances ${\displaystyle (b-a),}$ and ${\displaystyle (c-b),}$ are of very different sizes.

The modified-PERT distribution has been implemented in several simulation packages:

• ModelRisk [9] – risk analysis add-in for Excel.
• Primavera risk analysis – project risk analysis simulation tool.
• R (programming language) [10] - open-source open source programming language for statistical computing.
• Tamara [11] – project risk analysis simulation tool.
• Wolfram Mathematica [12] – mathematical symbolic computation program.

## Related Research Articles

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution, Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution is the distribution of the x-intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

In probability and statistics, Student's t-distribution is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym Student.

In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution.

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.

In probability theory and statistics, the Gumbel distribution is used to model the distribution of the maximum of a number of samples of various distributions.

In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution need not exist: this requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.

In probability theory and directional statistics, the von Mises distribution is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation. The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises–Fisher distribution on the N-dimensional sphere.

As with other probability distributions with noncentrality parameters, the noncentral t-distribution generalizes a probability distribution – Student's t-distribution – using a noncentrality parameter. Whereas the central distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false. This leads to its use in statistics, especially calculating statistical power. The noncentral t-distribution is also known as the singly noncentral t-distribution, and in addition to its primary use in statistical inference, is also used in robust modeling for data.

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above. The truncated normal distribution has wide applications in statistics and econometrics. For example, it is used to model the probabilities of the binary outcomes in the probit model and to model censored data in the Tobit model.

The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. There are several alternative formulations of this distribution in the literature. It is named after Z. W. Birnbaum and S. C. Saunders.

The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution. It has also been called the generalized logistic distribution, but this conflicts with other uses of the term: see generalized logistic distribution.

The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To distinguish the two families, they are referred to below as "version 1" and "version 2". However this is not a standard nomenclature.

In probability and statistics, the K-distribution is a three-parameter family of continuous probability distributions. The distribution arises by compounding two gamma distributions. In each case, a re-parametrization of the usual form of the family of gamma distributions is used, such that the parameters are:

In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.

In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle.

The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.

In probability theory, an exponentially modified Gaussian (EMG) distribution describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ2, and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component.

In probability theory and statistics, the beta rectangular distribution is a probability distribution that is a finite mixture distribution of the beta distribution and the continuous uniform distribution. The support is of the distribution is indicated by the parameters a and b, which are the minimum and maximum values respectively. The distribution provides an alternative to the beta distribution such that it allows more density to be placed at the extremes of the bounded interval of support. Thus it is a bounded distribution that allows for outliers to have a greater chance of occurring than does the beta distribution.

In probability theory and statistics, the generalized multivariate log-gamma (G-MVLG) distribution is a multivariate distribution introduced by Demirhan and Hamurkaroglu in 2011. The G-MVLG is a flexible distribution. Skewness and kurtosis are well controlled by the parameters of the distribution. This enables one to control dispersion of the distribution. Because of this property, the distribution is effectively used as a joint prior distribution in Bayesian analysis, especially when the likelihood is not from the location-scale family of distributions such as normal distribution.

In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables, inverse distributions are special cases of the class of ratio distributions, in which the numerator random variable has a degenerate distribution.

## References

1. Clark CE (1962) The PERT model for the distribution of an activity. Operations Research 10, pp. 405406
2. "PERT distribution". Vose Software. 2017-05-02. Retrieved 2017-07-16.
3. Continuous Univariate Distributions - 2nd Ed (1995). Johnson K, Kotz S and Balakkrishnan N. (Section 25.4)
4. Project Management Body of Knowledge: 5th Ed (2013). Project Management Institute Chapter 6
5. Simulation Modeling and Analysis (2000). Law AM and Kelton WD. Section 6.11
6. Business Risk and Simulation Modelling in Practice (2015). M Rees. Section 9.1.8
7. Risk Analysis – a Quantitative Guide: 3rd Ed. (2008) Vose D
8. Paulo Buchsbaum (June 9, 2012). "Modified Pert Simulation" (PDF). Greatsolutions.com.br. Archived from the original on December 23, 2018. Retrieved July 14, 2017.
9. "Modified PERT distribution". Vose Software. 2017-05-02. Retrieved 2017-07-16.
10. "Probability distributions used in Tamara". Vose Software. 2017-05-02. Retrieved 2017-07-16.
11. "PERTDistribution—Wolfram Language Documentation". Reference.wolfram.com. Retrieved 2017-07-16.